and the differences between values of $\displaystyle f(x)$ for consecutive integer values of $\displaystyle x$ don't form any obvious pattern.

However (using a spreadsheet to do the number-crunching) I have found that for all the values of $\displaystyle n$ I've tried so far (which is up to $\displaystyle n = 7$), $\displaystyle f(2^n) = f(2^{n-1}) + 1$. Which is what made me think it might be worth looking at powers of $\displaystyle 2$ or the binary representation of $\displaystyle x$.

Further thoughts: as $\displaystyle x$ increases through consecutive integer values:

the value of the first term, $\displaystyle \lfloor\frac {x + 1}{2}\rfloor$ increases by $\displaystyle 1$ every second value of $\displaystyle x$; i.e. when $\displaystyle x =2, 4, 6, 8, ...$